Wave propagation in monatomic lattices with anharmonic potential

Whitham’s variational method for nonlinear dispersive wave propagation is used to obtain the nonlinear dispersion relation between frequency, wavenumber, and amplitude in a monatomic chain of particles. The interactive force in the chain is of third degree in the relative displacement of nearest neighbours. The frequency is shown to be larger than its infinitesimal amplitude value at low wavenumbers and less at high wavenumbers, changing sign at a critical frequency which depends on the force constants of the lattice. The periodic dependence of frequency upon wavenumber is preserved, but the cutoff frequency diminishes with increasing amplitude. The dilatation or expansion of the lattice is found always to be positive and it increases with wavenumber and amplitude. The Gruneisen ratio, i. e. the rela­tive rate of change of frequency with respect to dilatation, depends on wavenumber and force constants. The formula reproduces the value given by quasilinear theory at the edge of the Brillouin zone, but predicts a sharp decrease with decreasing wavenumber, leading even to negative ratios under some circumstances. Finally, the space-time evolution of modulation perturbations on the nonlinear uniform wave is analyzed on the basis of Karpman’s and Krushkal’s theories. The linear chain is found to be mechanically unstable for wavenumbers less than a wavenumber K0 defined by the wavenumber k0 of the uniform wave and the force constants, but the instability may or may not be severe, since the amplitude of the perturbation is self-limiting.